False statementConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-08-31

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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A regular value need not belong to the image

Statement

False claim: every regular value of a smooth map lies in its image.

Facts & Assumptions

Given: The map $f : R \to R$ , $f (x) = e^{x}$ .

[F1]

A value is regular when every point of its fibre is regular, and the empty fibre is allowed (Regular and critical points and values).

[L1]

The derivative of $e^{x}$ is $e^{x}$ (The exponential function is smooth and $(\exp)^{'} = \exp$ ), and the exponential maps $R$ bijectively onto $(0, \infty)$ (The exponential is a continuous bijection from $R$ onto $(0, \infty)$ ).

Refutation

technique · direct

1.1

The value $0$ is not in the image of $f$ , because $e^{x} > 0$ for every real $x$ .

given

2.1

The fibre $f^{- 1} (0)$ is empty. By [F1], the regular-value condition is therefore vacuous, and [L1] is consistent with that since there are no nonregular fibre points to check.

F1L1step 1.1

3.1

Thus $0$ is a regular value that is not attained, so the claim is false.

step 1.1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

14 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Nigel Hitchin, Differentiable Manifolds, Theorem 3.3 context (standard reference, not scraped)